From 95423ec77a7a15bd7d0f7213946b5bb8bb819d0e Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Thu, 12 Sep 2013 16:14:33 +0200 Subject: [PATCH] fixed typo --- documents/Numerik/Klausur1/Aufgabe2.tex | 48 +++++++++++++++--------- documents/Numerik/Klausur1/Klausur1.pdf | Bin 193428 -> 200251 bytes 2 files changed, 30 insertions(+), 18 deletions(-) diff --git a/documents/Numerik/Klausur1/Aufgabe2.tex b/documents/Numerik/Klausur1/Aufgabe2.tex index 2d8376d..0a681bf 100644 --- a/documents/Numerik/Klausur1/Aufgabe2.tex +++ b/documents/Numerik/Klausur1/Aufgabe2.tex @@ -8,28 +8,40 @@ wobei $L$ eine invertierbare, untere Dreiecksmatrix ist. Geben Sie die Formel zur Berechnung von $y_i$ an. \textbf{Lösung:} +\[y_i = \frac{b_i - \sum_{k=1}^{i-1} l_{ik} \cdot y_k}{l_{ii}}\] -\[y_i = \frac{b_i - \sum_{k=i}^{i-1} l_{ik} \cdot y_k}{l_{ii}}\] +\begin{algorithm}[H] + \begin{algorithmic} + \Require Lower, invertable, triangular Matrix $L \in \mathbb{R}^{n \times n}$, Vektor $b$ + \Procedure{solve}{$L$, $b$} + \For{$i \in \Set{1, \dots n}$} + \State $y_i \gets b_i$ + \For{$k \in \Set{1, \dots, i-1}$} + \State $y_i \gets y_i - l_{ik} \cdot y_k$ + \EndFor + \State $y_i \gets \frac{y_i}{l_{ii}}$ + \EndFor + \EndProcedure + \end{algorithmic} +\caption{Calculate $y$ in $Ly = b$} +\end{algorithm} \subsection*{Teilaufgabe b} -\[Ax = b ? PAx = Pb ? LRx = Pb \] +\[Ax = b \Leftrightarrow PAx = Pb \Leftrightarrow LRx = Pb \] -Pseudocode: - - \begin{algorithm}[H] - \begin{algorithmic} - \Require Matrix $A$, Vektor $b$ - \Procedure{CalculateLegendre}{$A$, $b$} - \State $P, L, R \gets \Call{LRZerlegung}{A}$ - \State $b^* \gets Pb$ - \State $c \gets \Call{VorwärtsSubstitution}{L, b^*}$ - \State $x \gets \Call{RückwärtsSubstitution}{R, c}$ - \State \Return $x$ - \EndProcedure - \end{algorithmic} - \caption{Calculate TODO} - \label{alg:TODO} - \end{algorithm} +\begin{algorithm}[H] + \begin{algorithmic} + \Require Matrix $A$, Vektor $b$ + \Procedure{LoeseLGS}{$A$, $b$} + \State $P, L, R \gets \Call{LRZer}{A}$ + \State $b^* \gets Pb$ + \State $c \gets \Call{VorSub}{L, b^*}$ + \State $x \gets \Call{RueckSub}{R, c}$ + \State \Return $x$ + \EndProcedure + \end{algorithmic} +\caption{Löse ein LGS $Ax = b$} +\end{algorithm} \subsection*{Teilaufgabe c} Der Gesamtaufwand ist: diff --git a/documents/Numerik/Klausur1/Klausur1.pdf 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